3.74 \(\int \tan ^4(y) \, dy\)

Optimal. Leaf size=14 \[ y+\frac {\tan ^3(y)}{3}-\tan (y) \]

[Out]

y-tan(y)+1/3*tan(y)^3

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ y+\frac {\tan ^3(y)}{3}-\tan (y) \]

Antiderivative was successfully verified.

[In]

Int[Tan[y]^4,y]

[Out]

y - Tan[y] + Tan[y]^3/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

\begin {align*} \int \tan ^4(y) \, dy &=\frac {\tan ^3(y)}{3}-\int \tan ^2(y) \, dy\\ &=-\tan (y)+\frac {\tan ^3(y)}{3}+\int 1 \, dy\\ &=y-\tan (y)+\frac {\tan ^3(y)}{3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.29 \[ y-\frac {4 \tan (y)}{3}+\frac {1}{3} \tan (y) \sec ^2(y) \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[y]^4,y]

[Out]

y - (4*Tan[y])/3 + (Sec[y]^2*Tan[y])/3

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fricas [B]  time = 0.45, size = 26, normalized size = 1.86 \[ \frac {3 \, y \cos \relax (y)^{3} - {\left (4 \, \cos \relax (y)^{2} - 1\right )} \sin \relax (y)}{3 \, \cos \relax (y)^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)^4/cos(y)^4,y, algorithm="fricas")

[Out]

1/3*(3*y*cos(y)^3 - (4*cos(y)^2 - 1)*sin(y))/cos(y)^3

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giac [A]  time = 0.99, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (y)^{3} + y - \tan \relax (y) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)^4/cos(y)^4,y, algorithm="giac")

[Out]

1/3*tan(y)^3 + y - tan(y)

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maple [A]  time = 0.02, size = 13, normalized size = 0.93 \[ \frac {\left (\tan ^{3}\relax (y )\right )}{3}+y -\tan \relax (y ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)^4/cos(y)^4,y)

[Out]

y-tan(y)+1/3*tan(y)^3

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maxima [A]  time = 1.33, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (y)^{3} + y - \tan \relax (y) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)^4/cos(y)^4,y, algorithm="maxima")

[Out]

1/3*tan(y)^3 + y - tan(y)

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mupad [B]  time = 0.07, size = 12, normalized size = 0.86 \[ \frac {{\mathrm {tan}\relax (y)}^3}{3}-\mathrm {tan}\relax (y)+y \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(y)^4/cos(y)^4,y)

[Out]

y - tan(y) + tan(y)^3/3

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sympy [A]  time = 0.07, size = 19, normalized size = 1.36 \[ y + \frac {\sin ^{3}{\relax (y )}}{3 \cos ^{3}{\relax (y )}} - \frac {\sin {\relax (y )}}{\cos {\relax (y )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(y)**4/cos(y)**4,y)

[Out]

y + sin(y)**3/(3*cos(y)**3) - sin(y)/cos(y)

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